abraham robinson and nonstandard analysis: history ...mcps.umn.edu/philosophy/11_7dauben.pdf ·...

Joseph W Dauben

Abraham Robinson and Nonstandard

Analysis: History, Philosophy, and

Foundations of Mathematics

Mathematics is the subject in which we don't know* whatwe are talking about.

—Bertrand Russell

* Don't care would be more to the point.—Martin Davis

I never understood why logic should be reliableeverywhere else, but not in mathematics.

—A. Heyting

1. Infinitesimals and the History of MathematicsHistorically, the dual concepts of infinitesimals and infinities have

always been at the center of crises and foundations in mathematics, fromthe first "foundational crisis" that some, at least, have associated withdiscovery of irrational numbers (more properly speaking, incommensurablemagnitudes) by the pre-socartic Pythagoreans1, to the debates that are cur-rently waged between intuitionists and formalist—between the descendantsof Kronecker and Brouwer on the one hand, and of Cantor and Hilberton the other. Recently, a new "crisis" has been identified by the construc-tivist Erret Bishop:

There is a crisis in contemporary mathematics, and anybody who has

This paper was first presented as the second of two Harvard Lectures on Robinson and hiswork delivered at Yale University on 7 May 1982. In revised versions, it has been presentedto colleagues at the Boston Colloquim for the Philosophy of Science (27 April 1982), theAmerican Mathematical Society meeting in Chicago (23 March 1985), the Conference onHistory and Philosophy of Modern Mathematics held at the University of Minnesota (17-19May 1985), and, most recently, at the Centre National de Recherche Scientifique in Paris(4 June 1985) and the Department of Mathematics at the University of Strassbourg (7 June1985). I am grateful for energetic and constructive discussions with many colleagues whosecomments and suggestions have served to develop and sharpen the arguments presented here.

777

775 Joseph W. Dauben

not noticed it is being willfully blind. The crisis is due to our neglectof philosophical issues... .2

Bishop, too, relates his crisis in part to the subject of the infinite and in-finitesimals. Arguing that formalists mistakenly concentrate on the "truth"rather than the "meaning" of a mathematical statement, he criticizesAbraham Robinson's nonstandard analysis as "formal finesse," addingthat "it is difficult to believe that debasement of meaning could be car-ried so far."3 Not all mathematicians, however, are prepared to agree thatthere is a crisis in modern mathematics, or that Robinson's work con-stitutes any debasement of meaning at all.

Kurt Godel, for example, believed that Robinson more than anyoneelse had succeeded in bringing mathematics and logic together, and hepraised Robinson's creation of nonstandard analysis for enlisting thetechniques of modern logic to provide rigorous foundations for the calculususing actual infinitesimals. The new theory was first given wide publicityin 1961 when Robinson outlined the basic idea of his "nonstandard"analysis in a paper presented at a joint meeting of the AmericanMathematical Society and the Mathematical Association of America.4

Subsequently, impressive applications of Robinson's approach to infin-itesimals have confirmed his hopes that nonstandard analysis could enrich"standard" mathematics in important ways.

As for his success in defining infinitesimals in a rigorously mathematicalway, Robinson saw his work not only in the tradition of others like Leib-niz and Cauchy before him, but even as vindicating and justifying theirviews. The relation of their work, however, to Robinson's own researchis equally significant, as Robinson himself realized, and this for reasonsthat are of particular interest to the historian of mathematics. Beforereturning to the question of a "new" crisis in mathematics due to Robin-son's work, it is important to say something, briefly, about the historyof infinitesimals, a history that Robinson took with the utmost seriousness.

This is not the place to rehearse the long history of infinitesimals inmathematics. There is one historical figure, however, who especiallyinterested Robinson—namely, Cauchy—and in what follows Cauchy pro-vides a focus for considering the historiographic significance of Robin-son's own work. In fact, following Robinson's lead, others like J. P.Cleave, Charles Edwards, Detlef Laugwitz, and W. A. J. Luxemburg haveused nonstandard analysis to rehabilitate or "vindicate" earlier infin-itesimalists.5 Leibniz, Euler, and Cauchy are among the more promi-

ABRAHAM ROBINSON AND NONSTANDARD ANALYSIS 179

nent mathematicians who have been rationally reconstructed—even to thepoint of having had, in the views of some commentators, "Robinsonian"nonstandard infinitesimals in mind from the beginning. The most detailedand methodically sophisticated of such treatments to date is that providedby Imre Lakatos; in what follows, it is his analysis of Cauchy that isemphasized.

2. Lakatos, Robinson, and Nonstandard Interpretations ofCauchy's Infinitesimal Calculus

In 1966, Lakatos read a paper that provoked considerable discussionat the International Logic Colloquium meeting that year in Hannover.The primary aim of Lakatos's paper was made clear in its title: "Cauchyand the Continuum: The Significance of Non-standard Analysis for theHistory and Philosophy of Mathematics."6 Lakatos acknowledged his ex-changes with Robinson on the subject of nonstandard analysis as he con-tinued to revise the working draft of his paper. Although Lakatos neverpublished the article, it enjoyed a rather wide private circulation and even-tually appeared after Lakatos's death in volume 2 of his Mathematics,Science and Epistemology.

Lakatos realized that two important things had happened with the ap-pearance of Robinson's new theory, indebted as it was to the results andtechniques of modern mathematical logic. He took it above all as a signthat metamathematics was turning away from its original philosophicalbeginnings and was growing into an important branch of mathematics.7

This view, now more than twenty years later, seems fully justified.The second claim that Lakatos made, however, is that nonstandard

analysis revolutionizes the historian's picture of the history of the calculus.The grounds for this assertion are less clear—and in fact, are subject toquestion. Lakatos explained his interpretation of Robinson's achievementas follows at the beginning of his paper:

Robinson's work . . . offers a rational reconstruction of the discreditedinfinitesimal theory which satisfies modern requirements of rigour andwhich is no weaker than Weierstrass's theory. This reconstruction makesinfinitesimal theory an almost respectable ancestor of a fully-fledged,powerful modern theory, lifts it from the status of pre-scientific gib-berish and renews interest in its partly forgotten, partly falsified history.8

But consider the word almost. Robinson, says Lakatos, only makesthe achievements of earlier infinitesimalists almost respectable. In fact,


Robinson's work in the twentieth century cannot vindicate Leibniz's workin the seventeenth century, Euler's in the eighteenth century, or Cauchy'sin the nineteenth century. There is nothing in the language or thought ofLeibniz, Euler, or Cauchy (to whom Lakatos devotes most of his atten-tion) that would make them early Robinsonians. The difficulties ofLakatos's rational reconstruction, however, are clearer in some of thedetails he offers.

For example, consider Lakatos's interpretation of the famous theoremfrom Cauchy's Cours d'analyse of 1821, which purports to prove that thelimit of a sequence of continuous functions sn(x) is continuous. This is

what Lakatos, in the spirit of Robinson's own reading of Cauchy, hasto say:

In fact Cauchy's theorem was true and his proof as correct as an infor-mal proof can be. Following Robinson . . . Cauchy's argument, if notinterpreted as a proto-Weierstrassian argument but as a genuine Leibniz-Cauchy one, runs as follows:...sn (x) should be defined and continuous and converge not only at stan-dard Weierstrassian points but at every point of the "denser" Cauchycontinuum, and . . . the sequence sn(x) should be defined for infinitelylarge indices n and represent continuous functions at such indices.9

In one last sentence, this is all summarized in startling terms as follows:Cauchy made absolutely no mistake, he only proved a completely dif-ferent theorem, about transfinite sequences of functions which Cauchy-converge on the Leibniz continuum.10

But upon reading Cauchy's Cours d'analyse—or either of his laterpresentations of the theorem in his Resumes analytiques of 1833 or in theComptes Rendues for 1853—one finds no hint of transfinite indices, se-quences, or Leibnizian continua made "denser" than standard intervalsby the addition of infinitesimals. Cauchy, when referring to infinitely largenumbers n' > n, has "very large"—but finite—numbers in mind, not ac-tually infinite Cantorian-type transfinite numbers.11

This is unmistakably clear from another work Cauchy published in1833—Sept lecons dephysique generate—given at Turin in the same yearhe again published the continuous sum theorem. In the Sept lecons,however, Cauchy explicitly denies the existence of infinitely large numbersfor their allegedly contradictory properties.12

Moreover, if Lakatos was mistaken about Cauchy's position concern-ing the actually infinite, he was also wrong about Cauchy's continuum


being one of Leibnizian infinitesimals. If, by virtue of such infinitesimals,Cauchy's original proof had been correct all along, why would he thenhave issued a revised version in 1853, explicitly to improve upon the earlierproofs? Instead, were Lakatos and Robinson correct in their rationalreconstructions, all Cauchy would need to have done was point out thenonstandard meaning of his infinitesimals—explaining how infinitely largeand infinitely small numbers had given him a correct theorem, as well asa proof, all along.

Lakatos also draws some rather remarkable conclusions about why theLeibnizian version of nonstandard analysis failed:

The downfall of Leibnizian theory was then not due to the fact thatit was inconsistent, but that it was capable only of limited growth. Itwas the heuristic potential of growth—and explanatory power—ofWeierstrass's theory that brought about the downfall of infinitesimals.13

This rational reconstruction may complement the overall view Lakatostakes of the importance of research programs in the history of science,but it does no justice to Leibniz or to the subsequent history of the calculusin the eighteenth and early nineteenth centuries, which (contrary toLakatos) demonstrates that (i) in the eighteenth century the (basically Leib-nizian) calculus constituted a theory of considerable power in the handsof the Bernoullis, Euler, and many others; and (ii) the real stumbling blockto infinitesimals was their acknowledged inconsistency.

The first point is easily established by virtue of the remarkable achieve-ments of eighteenth-century mathematicians who used the calculus becauseit was powerful—it produced striking results and was indispensable in ap-plications.14 But it was also suspect from the beginning, and preciselybecause of the question of the contradictory nature of infinitesimals.

This brings us to the second point: despite Lakatos's dismissal of theirinconsistency, infinitesimals were perceived even by Newton and Leibniz,and certainly by their successors in the eighteenth century, as problematicprecisely because of their contradictory qualities. Newton was specifical-ly concerned with the fact that infinitesimals did not obey the Archime-dean axiom and therefore could not be accepted as part of rigorousmathematics.15 Leibniz was similarly concerned about the logical accept-ability of infinitesimals. The first public presentation of his differentialcalculus in 1684 was severely determined by his attempt to avoid the logicaldifficulties connected with the infinitely small. His article in the ActaEruditorum on maxima and minima, for example, presented the differen-


tial as a finite line segment rather than the infinitely small quantity thatwas used in practice.16

This confusion between theoretical considerations and practical applica-tions carried over to Leibniz's metaphysics of the infinite, for he was nevercommitted to any one view but made conflicting pronouncements.Philosophically, as Robinson himself has argued, Leibniz had to assumethe reality of the infinite—the infinity of his monads, for example—orthe reality of infinitesimals not as mathematical points but as substance-or force-points—namely, Leibniz's "monads" themselves.17

That the eighteenth century was concerned not with doubts about thepotential of infinitesimals but primarily with fears about their logical con-sistency is clear from the proposal Lagrange drew up for a prize to beawarded by the Berlin Academy for a rigorous theory of infinitesimals.As the prize proposal put it:

It is well known that higher mathematics continually uses infinitely largeand infinitely small quantities. Nevertheless, geometers, and even theancient analysts, have carefully avoided everything which approachesthe infinite; and some great modern analysts hold that the terms of theexpression infinite magnitude contradict one another.The Academy hopes, therefore, that it can be explained how so manytrue theorems have been deduced from a contradictory supposition,and that a principle can be delineated which is sure, clear—in a word,truly mathematical—which can appropriately be substituted for theinfinite.18

Lakatos seems to appreciate all this—and even contradicts himself onthe subject of Leibniz's theory and the significance of its perceived in-consistency. Recalling his earlier assertion that Leibniz's theory was notoverthrown because of its inconsistency, consider the following line, justa few pages later, where Lakatos asserts that nonstandard analysis raisesthe problem of "how to appraise inconsistent theories like Leibniz'scalculus, Frege's logic, and Dirac's delta function."19

Lakatos apparently had not made up his mind as to the significanceof the inconsistency of Leibniz's theory, which raises questions about thehistorical value and appropriateness of the extreme sort of rationalreconstruction that he has proposed to "vindicate" the work of earliergenerations. In fact, neither Leibniz nor Euler nor Cauchy succeeded ingiving a satisfactory foundation for an infinitesimal calculus that alsodemonstrated its logical consistency. Basically, Cauchy's "epsilontics"


were a means of avoiding infinites and infinitesimals. Nowhere do Robin-sonian infinitesimals or justifications appear in Cauchy's explanations ofthe rigorous acceptability of his work.20

Wholly apart from what Lakatos and others like Robinson have at-tempted in reinterpreting earlier results in terms of nonstandard analysis,it is still important to understand Robinson's own reasons for developinghis historical knowledge in as much detail—and with as much scholar-ship—as he did. For Robinson, the history of infinitesimals was more thanan antiquarian interest; it was not one that developed with advancing ageor retirement, but was a simultaneous development that began with hisdiscovery of nonstandard analysis in the early 1960s. Moreover, there seemto have been serious reasons for Robinson's keen attention to the historyof mathematics as part of his own "research program" concerned withthe future of nonstandard analysis.

3. Nonstandard Analysis and the History of Mathematics

In 1965, in a paper titled "On the Theory of Normal Families," Robin-son began with a short look at the history of mathematics.21 He notedthat for about one hundred and fifty years after its inception in the seven-teenth century, mathematical analysis developed vigorously on inadequatefoundations. Despite this inadequacy, the precise, quantitative results pro-duced by the leading mathematicians of that period have stood the testof time.

In the first half of the nineteenth century, however, the concept of thelimit, advocated previously by Newton and d'Alembert, gained ascendan-cy. Cauchy, whose influence was instrumental in bringing about thechange, still based his arguments on the intuitive concept of an infinitelysmall number as a variable tending to zero. At the same time, however,he set the stage for the formally more satisfactory theory of Weierstrass,and today deltas and epsilons are the everyday language of the calculus,at least for most mathematicians. It was this precise approach that pavedthe way for the formulation of more general and more abstract concepts.Robinson used this history to explain the importance of compactness asapplied to functions of a complex variable, which had led to the theoryof normal families developed largely by Paul Montel. There followed thequalitative development of complex variable theory, such as Picard theory,and, finally, against this background, more quantitative theories like thosedeveloped by Rolf Nevanlinna—to whom Robinson's paper was dedicatedas part of a Festschrift.


The historical notes to be found at the beginning of Robinson's paperwere echoed again at the end, when he turned to ask whether the resultshe had achieved using nonstandard analysis couldn't be achieved justas well by standard methods. Although he admitted that because of thetransfer principle (developed in his paper of 1961, "Non-Standard Anal-ysis") this was indeed possible, he added that such translations into stand-ard terms usually complicated matters considerably. As for nonstandardanalysis and the use of infinitesimals it permitted, his conclusion wasemphatic:

Nevertheless, we venture to suggest that our approach has a certainnatural appeal, as shown by the fact that it was preceded in historyby a long line of attempts to introduce infinitely small and infinitelylarge numbers into Analysis.22

And so the reason for the historical digression was its usefulness in serv-ing a much broader purpose than merely introducing some rather remotehistorical connections between Newton, Leibniz, Paul Montel, and RolfNevanlinna. History could serve the mathematician as propaganda. Robin-son was apparently concerned that many mathematicians were preparedto adopt a "so what" attitude toward nonstandard analysis because ofthe more familiar reduction that was always possible to classical founda-tions. There were several ways to outflank those who chose to minimizenonstandard analysis because, theoretically, it could do nothing that wasn'tequally possible in standard analysis. Above all, nonstandard analysis wasoften simpler and more intuitive in a very direct, immediate way thanstandard approaches. But, as Robinson also began to argue with increas-ing frequency and in greater detail, historically the concept of infinitesimalshad always seemed natural and intuitively preferable to more convolutedand less intuitive sorts of rigor. Now that nonstandard analysis showedwhy infinitesimals were safe for consumption in mathematics, there wasno reason not to exploit their natural advantages. The paper for RolfNevanlinna was meant to exhibit both the technical applications and, atleast in part through its appeal to history, the naturalness of nonstandardanalysis in developing the theory of normal families.

4. Foundations and Philosophy of Mathematics

If Robinson regarded the history of infinitesimals as an aid to thejustification in a very general way of nonstandard analysis, what contribu-tion did it make, along with his results in model theory, to the founda-


tions and philosophy of mathematics? Stephan Korner, who taught aphilosophy of mathematics course with Robinson at Yale in the fall of1973, shortly before Robinson's death early the following year, wasdoubtless closest to Robinson's maturest views on the subject.23 Basical-ly, Korner sees Robinson as a follower of—or at least working in the samespirit as—Leibniz and Hilbert. Like Leibniz (and Kant after him), Robin-son rejected any empirical basis for knowledge about the infinite—whetherin the form of infinitely large or infinitely small quantities, sets, whatever.Leibniz is famous for his view that infinitesimals are useful fictions—aposition deplored by such critics as Nieuwentijt or the more flamboyantand popular Bishop Berkeley, whose condemnation of the Newtoniancalculus might equally well have applied to Leibniz.24 Leibniz adopted boththe infinitely large and the infinitely small in mathematics for pragmaticreasons, as permitting an economy of expression and an intuitive, sug-gestive, heuristic picture. Ultimately, there was nothing to worry aboutsince the mathematician could eliminate them from his final result afterhaving infinitesimals and infinities to provide the machinery and do thework of a proof.

Leibniz and Robinson shared a similar view of the ontological statusof infinities and infinitesimals. They are not just fictions, but well-foundones—"fictiones bene fondatae," in the sense that their applications proveuseful in penetrating the complexity of natural phenomena and help toreveal relationships in nature that purely empirical investigations wouldnever produce.

As Emil Borel once said of Georg Cantor's transfinite set theory (toparaphrase not too grossly): although he objected to transfinite numbersor inductions in the formal presentation of finished results, it was cer-tainly permissible to use them to discover theorems and create proofs—again, whatever works.25 It was only necessary to be sure that in the finalversion they were eliminated, thus making no official appearance. Robin-son, however, was interested in more, especially in the reasons why themathematics worked as it did, and in particular why infinities and in-finitesimals were now admissible as rigorous entities despite centuries ofdoubts and attempts to eradicate them entirely.

Here Robinson succeeded where Leibniz and his successors failed. Leib-niz, for example, never demonstrated the consistent foundations of hiscalculus, for which his work was sharply criticized by Nieuwentijt, amongothers. Throughout the eighteenth century, the troubling foundations (real-


ly, lack of foundations) of the Leibnizian infinitesmal calculus continuedto bother mathematicians, until the epsilon-delta methods of Cauchy andthe "arithmetic" rigor of Weierstrass reestablished analysis on acceptablyfinite terms. Because, as Korner remarks, "Leibniz's approach was con-sidered irremediably inconsistent, hardly any efforts were made to improvethis delimination."26

Robinson was clearly not convinced of the inconsistency of infini-tesimals, and in developing the methods of Skolem (who had advancedthe idea of nonstandard arithmetic) he was led to consider the possibilityof nonstandard analysis. At the same time, his work in model theory andmathematical logic contributed not only to his creation of nonstandardanalysis, but to his views on the foundations of mathematics as well.

5. Robinson and "Formalism 64"

In the 1950s, working under the influence of his teacher AbrahamFraenkel, Robinson seems to have been satisfied with a fairly straightfor-ward philosophy of Platonic realism. But by 1964, Robinson's philosoph-ical views had undergone considerable change. In a paper titled simply"Formalism 64," Robinson emphasized two factors in rejecting his earlierPlatonism in favor of a formalist position:

(i) Infinite totalities do not exist in any sense of the word (i.e., eitherreally or ideally). More precisely, any mention, or purported mention,of infinite totalities is, literally, meaningless.(ii) Nevertheless, we should continue the business of Mathematics "asusual," i.e. we should act as //infinite totalities really existed.27

Georg Kreisel once commented that, as he read Robinson's "Formalism64," it was not clear to him whether Robinson meant 1864 or 7P64! Robin-son, however, was clearly responding in his views on formalism to researchthat had made a startling impression upon mathematicians only in theprevious year—namely, Paul Cohen's important work in 1963 on forcingand the independence of the continuum hypothesis.

As long as it appeared that the accepted axiomatic systems of set theory(the Zermelo-Fraenkel axiomatization, for example) were able to cope withall set theoretical problems that were of interest to the working mathemati-cian, belief in the existence of a unique "universe of sets" was almostunanimous. However, this simple view of the situation was severely shakenin the 1950s and early 1960s by two distinct developments. One of thesewas Cohen's proof of the independence of the continuum hypothesis,


which revealed a great disparity between the scale of transfinite ordinalsand the scale of cardinals—or power sets. As Robinson himself noted inan article in Dialectica, the relation "is so flexible that it seems to be quitebeyond control, at least for now."28

The second development of concern to Robinson was the emergenceof new and varied axioms of infinity. Although the orthodox Platonistbelieves that in the real world such axioms must either be true or false,Robinson found himself persuaded otherwise. Despite his new approachto foundations in "Formalism 64," he was not dogmatic, but remainedflexible:

The development of "meaningless" infinitistic theories may at somefuture date become so unsatisfactory to me that I shall be willing toacknowledge the greater intellectual seriousness of some form of con-structivism. But I cannot imagine that I shall ever return to the creedof the true platonist, who sees the world of the actual infinite spreadout before him and believes that he can comprehend the incomprehen-sible.29

6. Erret Bishop: Meaning, Truth, and Nonstandard AnalysisIncomprehensible, however, is what some of Robinson's critics have

said, almost literally, of nonstandard analysis itself. Of all Robinson'sopponents, at least in public, none has been more vocal—or morevehement—than Erret Bishop.

In the summer of 1974, it was hoped that Robinson and Bishop wouldactually have a chance to discuss their views in a forum of mathemati-cians and historians and philosophers of mathematics who were invitedto a special Workshop on the Evolution of Modern Mathematics held atthe American Academy of Arts and Sciences in Boston. Garrett Birkhoff,one of the workshop's organizers, had intended to feature Robinson asthe keynote speaker for the section of the Academy's program devotedto foundations of mathematics, but Robinson's unexpected death in Aprilof 1974 made this impossible. Instead, Erret Bishop presented the featuredpaper for the section on foundations. Birkhoff compared Robinson's ideaswith those of Bishop in the following terms:

During the past twenty years, significant contributions to the founda-tions of mathematics have been made by two opposing schools. One,led by Abraham Robinson, claims Leibnizian antecedents for a "non-standard analysis" stemming from the "model theory" of Tarski. Theother (smaller) school, led by Errett Bishop, attempts to reinterpret


Brouwer's "intuitionism" in terms of concepts of "constructiveanalysis."30

Birkhoff went on to describe briefly (in a written report of the session)the spirited discussions following Bishop's talk, marked, as he noted, "bythe absence of positive reactions to Bishop's view."31 Even so, Bishop'spaper raised a fundamental question about the philosophy of mathematics,which he put simply as follows: "As pure mathematicians, we must decidewhether we are playing a game, or whether our theorems describe an ex-ternal reality."32 If these are the only choices, then one's response is ob-viously limited. For Robinson, the excluded middle would have to comeinto play here—for he viewed mathematics, in particular the striking resultshe had achieved in model theory and nonstandard analysis, as constitutingmuch more than a meaningless game, although he eventually came tobelieve that mathematics did not necessarily describe any external reality.But more of Robinson's own metaphysics in a moment.

Bishop made his concerns over the crisis he saw in contemporarymathematics quite clear in a dramatic characterization of what he tookto be the pernicious efforts of historians and philosophers alike. Not onlyis there a crisis at the foundations of mathematics, according to Bishop,but a very real danger (as he put it) in the role that historians seemed tobe playing, along with nonstandard analysis itself, in fueling the crisis:

I think that it should be a fundamental concern to the historians thatwhat they are doing is potentially dangerous. The superficial dangeris that it will be and in fact has been systematically distorted in orderto support the status quo. And there is a deeper danger: it is so easyto accept the problems that have historically been regarded as signifi-cant as actually being significant.33

Interestingly, in his own historical writing, Robinson sometimes madethe same point concerning the triumph, as many historians (and math-ematicians as well) have come to see it, of the success of Cauchy-Weier-strassian epsilontics over infinitesimals in making the calculus "rigorous"in the course of the nineteenth century. In fact, one of the most impor-tant achievements of Robinson's work in nonstandard analysis has beenhis conclusive demonstration of the poverty of this kind of historicism—of the mathematically Whiggish interpretation of increasing rigor over themathematically unjustifiable "cholera baccillus" of infinitesimals, to useGeorg Cantor's colorful description.34


As for nonstandard analysis, Bishop had this to say at the Bostonmeeting:

A more recent attempt at mathematics by formal finesse is nonstand-ard analysis. I gather that it has met with some degree of success,whether at the expense of giving significantly less meaningful proofsI do not know. My interest in nonstandard analysis is that attemptsare being made to introduce it into calculus courses. It is difficult tobelieve that debasement of meaning could be carried so far.35

Two things deserve comment here. The first is that Bishop (surprising-ly, in light of some of his later comments about nonstandard analysis)does not dismiss it as completely meaningless, but only asks whether itsproofs are "significantly less meaningful" than constructivist proofs. Leav-ing open for the moment what Bishop has in mind here for "meaningless"in terms of proofs, is seems clear that by one useful indicator to whichBishop refers, nonstandard analysis is year-by-year showing itself to beincreasingly "meaningful."36

Consider, for example, the pragmatic value of nonstandard analysisin terms of its application in teaching the calculus. Here it is necessaryto consider the success of Jerome Keisler's textbook Elementary Calculus:An Approach Using Infinitesimals, which uses nonstandard analysis toexplain in an introductory course the basic ideas of calculus. The issueof its pedagogic value will also serve to reintroduce, in a moment, the ques-tion of meaning in a very direct way.

Bishop claims that the use of nonstandard analysis to teach the calculusis wholly pernicious. He says this explicitly:

The technical complications introduced by Keisler's approach are ofminor importance. The real damage lies in his obfuscation anddevitalization of those wonderful ideas. No invocation of Newton andLeibniz is going to justify developing calculus using [nonstandardanalysis] on the grounds that the usual definition of a limit is toocomplicated!...Although it seems to be futile, I always tell my calculus students thatmathematics is not esoteric: it is commonsense. (Even the notoriouse, d definition of limit is commonsense, and moreover is central to theimportant practical problems of approximation and estimation.) Theydo not believe me.37

One reason Bishop's students may not believe him is that what he claims,


in fact, does not seem to be true. There is another side to this as well,for one may also ask whether there is any truth to the assertions madeby Robinson (and emphatically by Keisler) that "the whole point of ourinfinitesimal approach to calculus is that it is easier to define and explainlimits using infinitesimals."38 Of course, this claim also deserves examina-tion, in part because Bishop's own attempt to dismiss Keisler's methodsas being equivalent to the axiom "0 = 1" is simply nonsense.39 In fact,there are concrete indications that despite the allegations made by Bishopabout obfuscation and the nonintuitiveness of basic ideas in nonstandardterms, exactly the opposite is true.

Not long ago a study was undertaken to assess the validity of the claimthat "from this nonstandard approach, the definitions of the basic con-cepts [of the calculus] become simpler and the arguments more intuitive."40

Kathleen Sullivan reported the results of her dissertation, written at theUniversity of Wisconsin and designed to determine the pedagogicalusefulness of nonstandard analysis in teaching calculus, in the AmericanMathematical Monthly in 1976. This study, therefore, was presumablyavailable to Bishop when his review of Keisler's book appeared in 1977,in which he attacked the pedagogical validity of nonstandard analysis.What did Sullivan's study reveal? Basically, she set out to answer thefollowing questions:

Will the students acquire the basic calculus skills? Will they reallyunderstand the fundamental concepts any differently? How difficultwill it be for them to make the transition into standard analysis coursesif they want to study more mathematics? Is the nonstandard approachonly suitable for gifted mathematics students?41

To answer these questions, Sullivan studied classes at five schools inthe Chicago-Milwaukee area during the years 1973-74. Four of them weresmall private colleges, the fifth a public high school in a suburb ofMilwaukee. The same instructors who had taught the course previouslyagreed to teach one introductory course using Keisler's book (the 1971edition) as well as another introductory course using a standard approach(thus serving as a control group) to the calculus. Comparison of SAT scoresshowed that both the experimental (nonstandard) group and the standard(control) group were comparable in ability before the courses began. Atthe end of the course, a calculus test was given to both groups. Instruc-tors teaching the courses were interviewed, and a questionnaire was filledout by everyone who has used Keisler's book within the last five years.


The single question that brought out the greatest difference betweenthe two groups was question 3:

Define f(x) by the rule

Prove using the definition of limit that lim f(x) = 4.

Control Group Experimental Group(68 students) (68 students)

Did not attempt 22 4

Standard arguments:Satisfactory proof 2 14Correct statement, faulty proof 15 14Incorrect arguments 29 23

Nonstandard arguments:Satisfactory proof 25Incorrect arguments 2

The results, as shown in the accompanying tabulation, seem to be strik-ing; but, as Sullivan cautions:

Seeking to determine whether or not students really do perceive the basicconcepts any differently is not simply a matter of tabulating how manystudents can formulate proper mathematical definitions. Most teacherswould probably agree that this would be a very imperfect instrumentfor measuring understanding in a college freshman. But further lighton this and other questions can be sought in the comments of theinstructors.42

Here, too, the results are remarkable in their support of the heuristicvalue of using nonstandard analysis in the classroom. It would seem that,contrary to Bishop's views, the traditional approach to the calculus maybe the more pernicious. Instead, the new nonstandard approach waspraised in strong terms by those who actually used it:

The group as a whole responded in a way favorable to the experimen-tal method on every item: the students learned the basic concepts ofthe calculus more easily, proofs were easier to explain and closer tointuition, and most felt that the students end up with a better under-standing of the basic concepts of the calculus.43


As to Bishop's claim that the d,e method is "commonsense,"44 thistoo is open to question. As one teacher having successfully used Keisler'sbook remarked, "When my most recent classes were presented with theepsilon-delta definition of limit, they were outraged by its obscurity com-pared to what they had learned [via nonstandard analysis]."45

But as G. R. Blackley warned Keisler's publishers (Prindle, Weber andSchmidt) in a letter when he was asked to review the new textbook priorto its publication:

Such problems as might arise with the book will be political. It is revolu-tionary. Revolutions are seldom welcomed by the established party,although revolutionaries often are.46

The point to all of this is simply that, if one take meaning as the stan-dard, as Bishop urges, rather than truth, then it seems clear that by itsown success nonstandard analysis has indeed proven itself meaningful atthe most elementary level at which it could be introduced—namely, thatat which calculus is taught for the first time. But there is also a deeperlevel of meaning at which nonstandard analysis operates—one that alsotouches on some of Bishop's criticisms. Here again Bishop's views canalso be questioned and shown to be as unfounded as his objections tononstandard analysis pedagogically.

Recall that Bishop began his remarks in Boston at the AmericanAcademy of Arts and Sciences workshop in 1974 by stressing the crisisin contemporary mathematics that stemmed from what he perceived asa misplaced emphasis upon formal systems and a lack of distinction be-tween the ideas of "truth" and "meaning." The choice Bishop gave inBoston was between mathematics as a meaningless game or as a disciplinedescribing some objective reality. Leaving aside the question of whethermathematics actually describes reality, in some objective sense, considerRobinson's own hopes for nonstandard analysis, those beyond the pure-ly technical results he expected the theory to produce. In the preface tohis book on the subject, he hoped that "some branches of modernTheoretical Physics might benefit directly from the application of non-standard analysis."47

In fact, the practical advantages of using nonstandard analysis as abranch of applied mathematics have been considerable. Although this isnot the place to go into detail about the increasing number of results aris-ing from nonstandard analysis in diverse contexts, it suffices here to men-tion impressive research using nonstandard analysis in physics, especially


quantum theory and thermodynamics, and in economics, where study ofexchange economies has been particularly amenable to nonstandard in-terpretation.48

7. Conclusion

There is another purely theoretical context in which Robinson con-sidered the importance of the history of mathematics that also warrantsconsideration. In 1973, Robinson wrote an expository article that drewits title from a famous monograph written in the nineteenth century byRichard Dedekind: Was sind und was sollen die Zahlen? This title wasroughly translated—or transformed in Robinson's version—as"Numbers—What Are They and What Are They Good For?" As Robin-son put it: "Number systems, like hair styles, go in and out of fashion—it's what's underneath that counts."49

This might well be taken as the leitmotiv of much of Robinson'smathematical career, for his surpassing interest since the days of his disser-tation written at the University of London in the late 1940s was modeltheory, and especially the ways in which mathematical logic could not onlyilluminate mathematics, but have very real and useful applications withinvirtually all of its branches. In discussing number systems, he wanted todemonstrate, as he put it, that

the collection of all number systems is not a finished totality whosediscovery was complete around 1600, or 1700, or 1800, but that it hasbeen and still is a growing and changing area, sometimes absorbingnew systems and sometimes discarding old ones, or relegating them tothe attic.50

Robinson, of course, was leading up in his paper to the way in whichnonstandard analysis had again broken the bounds of the traditionalCantor-Dedekind understanding of the real numbers, especially as theyhad been augmented by Cantorian transfinite ordinals and cardinals.

To make his point, Robinson turned momentarily to the nineteenth cen-tury and noted that Hamilton had been the first to demonstrate that therewas a larger arithmetical system than that of the complex numbers—namely, that represented by his quaternions. These were soon supplantedby the system of vectors developed by Josiah Willard Gibbs of Yale andeventually transformed into a vector calculus. This was a more usefulsystem, one more advantageous in the sorts of applications for whichquaternions had been invented.


Somewhat later, another approach to the concept of number was takenby Georg Cantor, who used the idea of equinumerosity in terms of one-to-one correspondences to define numbers. In fact, for Cantor a cardinalnumber was a symbol assigned to a set, and the same symbol representedall sets equivalent to the base set. The advantage of this view of the natureof numbers, of course, was that it could be applied to infinite sets, pro-ducing transfinite numbers and eventually leading to an entire system oftransfinite arithmetic. Its major disadvantage, however, was that it ledCantor to reject adamantly any mathematical concept of infinitesimal.51

As Robinson points out, although the eventual fate of Cantor's theorywas a success story, it was not entirely so for its author. Despite the clearutility of Cantor's ideas, which arose in connection with his work ontrigonometric series (later applied with great success by Lebesgue andothers at the turn of the century), it was highly criticized by a spectrumof mathematicians, including, among the most prominent, Kronecker,Frege, and Poincare. In addition to the traditional objection that the in-finite should not be allowed in rigorous mathematics, Cantor's work wasalso questioned because of its abstract character. Ultimately, however,Cantor's ideas prevailed, despite criticism, and today set theory is a cor-nerstone, if not the major foundation, upon which much of modernmathematics rests.52

There was an important lesson to be learned, Robinson believed, inthe eventual acceptance of new ideas of number, despite their novelty orthe controversies they might provoke. Ultimately, utilitarian realities couldnot be overlooked or ignored forever. With an eye on the future of non-standard analysis, Robinson was impressed by the fate of another theorydevised late in the nineteenth century that also attempted, like those ofHamilton, Cantor, and Robinson, to develop and expand the frontiersof number.

In the 1890s, Kurt Hensel introduced a whole series of new numbersystems, his now familiar p-adic numbers. Hensel realized that he coulduse his p-adic numbers to investigate properties of the integers and othernumbers. He also realized, as did others, that the same results could beobtained in other ways. Consequently, many mathematicians came toregard Hensel's work as a pleasant game; but, as Robinson himself ob-served, "Many of Hensel's contemporaries were reluctant to acquire thetechniques involved in handling the new numbers and thought they con-stituted an unnecessary burden."53

The same might be said of nonstandard analysis, particularly in light


of the transfer principle that demonstrates that theorems true in *R canalso be proven for R by standard methods. Moreover, many mathemati-cians are clearly reluctant to master the logical machinery of model theorywith which Robinson developed his original version of nonstandardanalysis. This problem has been resolved by Keisler and Luxemburg,among others, who have presented nonstandard analysis in ways accessi-ble to mathematicians without their having to take up the difficulties ofmathematical logic as a prerequisite.54 But for those who see nonstandardanalysis as a fad that may be a currently pleasant game, like Hensel's p-adicnumbers, the later history of Hensel's ideas should give skeptics an ex-ample to ponder. For today, p-adic numbers are regarded as coequal withthe reals, and they have proven a fertile area of mathematical research.

The same has been demonstrated by nonstandard analysis. Its applica-tions in areas of analysis, the theory of complex variables, mathematicalphysics, economics, and a host of other fields have shown the utility ofRobinson's own extension of the number concept. Like Hensel's p-adicnumbers, nonstandard analysis can be avoided, although to do so maycomplicate proofs and render the basic features of an argument lessintuitive.

What pleased Robinson as much about nonstandard analysis as the in-terest it engendered from the beginning among mathematicians was theway it demonstrated the indispensability, as well as the power, of technicallogic:

It is interesting that a method which had been given up as untenablehas at last turned out to be workable and that this development in aconcrete branch of mathematics was brought about by the refined toolsmade available by modern mathematical logic.55

Robinson had begun his career as a mathematician by studying settheory and axiomatics with Abraham Fraenkel in Jerusalem, which even-tually led to his Ph.D. from the University of London in 1949.56 His earlyinterest in logic was later amply repaid in his applications of logic to thedevelopment of nonstandard analysis. As Simon Kochen once put it inassessing the significance of Robinson's contributions to mathematicallogic and model theory:

Robinson, via model theory, wedded logic to the mainstreams ofmathematics.... At present, principally because of the work of


Abraham Robinson, model theory is just that: a fully-fledged theorywith manifold interrelations with the rest of mathematics.57

Kurt Godel valued Robinson's achievement for similar reasons: it suc-ceeded in uniting mathematics and logic in an essential, fundamental way.That union has proved to be not only one of considerable mathematicalimportance, but of substantial philosophical and historical content as well.

Notes1. There is a considerable literature on the subject of the supposed crisis in mathematics

associated with the Pythagoreans. See, for example, (Hasse and Scholz 1928). For a recentsurvey of this debate, see (Berggren 1984; Dauben 1984; Knorr 1975).

2. (Bishop 1975, 507).3. (Bishop 1975, 513-14).4. Robinson first published the idea of nonstandard analysis in a paper submitted to

the Dutch Academy of Sciences (Robinson 1961).5. (Cleave 1971; Edwards 1979; Laugwitz 1975, 1985; Luxemburg 1975).6. (Lakatos 1978).7. (Lakatos 1978, 43).8. (Lakatos 1978, 44).9. (Lakatos 1978, 49).10. (Lakatos 1978, 50). Emphasis in original.11. Cauchy offers his definitions of infinitely large and small numbers in several works,

first in the Cours d'analyse, subsequently in later versions without substantive changes. See(Cauchy 1821, 19; 1823, 16; 1829, 265), as well as (Fisher 1978).

12. (Cauchy 1868).13. (Lakatos 1978, 54).14. For details of the successful development of the early calculus, see (Boyer 1939;

Grattan-Guinness 1970, 1980; Grabiner 1981; Youshkevitch 1959).15. (Newton 1727, 39), where he discusses the contrary nature of indivisibles as

demonstrated by Euclid in Book X of the Elements. For additional analysis of Newton'sviews on infinitesimals, see (Grabiner 1981, 32).

16. See (Leibniz 1684). For details and a critical analysis of what is involved in Leibniz'spresentation and applications of infinitesimals, see (Bos 1974-75; Engelsman 1984).

17. See (Robinson 1967, 35 [in Robinson 1979, 544]).18. In (Lagrange 1784, 12-13; Dugac 1980, 12). For details of the Berlin Academy's com-

petition, see (Grabiner 1981, 40-43; Youshkevitch 1971, 149-68).19. (Lakatos 1978, 59). Emphasis added.20. See (Grattan-Guinness 1970, 55-56), where he discusses "limit-avoidance" and its

role in making the calculus rigorous.21. (Robinson 1965b).22. (Robinson 1965b, 184); also in (Robinson 1979, vol. 2, 87).23. I am grateful to Stephan Korner and am happy to acknowledge his help in ongoing

discussions we have had of Robinson and his work.24. For a recent survey of the controversies surrounding the early development of the

calculus, see (Hall 1980).25. Borel in a letter to Hadamard, in (Borel 1928, 158).26. (KOrner 1979, xlii). Korner notes, however, that an exception to this generalization

is to be found in Hans Vaihinger's general theory of fictions. Vaihinger tried to justify in-finitesimals by "a method of opposite mistakes," a solution that was too imprecise, Kornersuggests, to have impressed mathematicians. See (Vaihinger 1913, 51 Iff).


27. (Robinson 1965a, 230; Robinson 1979, 507). Nearly ten years later, Robinson re-called the major points of "Formalism 64" as follows: "(i) that mathematical theories which,allegedly, deal with infinite totalities do not have any detailed meaning, i.e. reference, and(ii) that this has no bearing on the question whether or not such theories should be developedand that, indeed, there are good reasons why we should continue to do mathematics in theclassical fashion nevertheless." Robinson added that nothing since 1964 had prompted himto change these views and that, in fact, "well-known recent developments in set theory repre-sent evidence favoring these views." See (Robinson 1975, 557).

28. (Robinson 1970, 45-49).29. (Robinson 1970, 45-49).30. (Birkhoff 1975, 504).31. (Birkhoff 1975, 504).32. (Bishop 1975, 507).33. (Bishop 1975, 508).34. For Cantor's views, see his letter to the Italian mathematician Vivanti in (Meschkowski

1965, 505). A general analysis of Cantor's interpretation of infinitesimals may be foundin (Dauben 1979, 128-32, 233-38). On the question of rigor, see (Grabiner 1974).

35. (Bishop 1975, 514).36. It should also be noted, if only in passing, that Bishop has not bothered himself,

apparently, with a careful study of nonstandard analysis or its implications, for he offhandedlyadmits that he only "gathers that it has met with some degree of success" (Bishop 1975,514; emphasis added).

37. (Bishop 1977, 208).38. (Keisler 1976, 298), emphasis added; quoted in (Bishop 1977, 207).39. (Bishop 1976, 207).40. (Sullivan 1976, 370). Note that Sullivan's study used the experimental version of

Keisler's book, issued in 1971. Bishop reviewed the first edition published five years laterby Prindle, Weber and Schmidt. See (Keisler 1971, 1976).

41. (Sullivan 1976, 371).42. (Sullivan 1976, 373).43. (Sullivan 1976, 383-84).44. (Bishop 1977, 208).45. (Sullivan 1976, 373).46. (Sullivan 1976, 375).47. (Robinson 1966, 5).48. See especially (Robinson 1972a, 1972b, 1974, 1975), as well as (Dresden 1976) and

(Voros 1973).49. (Robinson 1973, 14).50. (Robinson 1973, 14).51. For details, see (Dauben 1979).52. See (Dauben 1979).53. (Robinson 1973, 16).54. (Luxemburg 1962, 1976; Keisler 1971).55. (Robinson 1973, 16).56. Robinson completed his dissertation, The Metamathematics of Algebraic Systems,

at Birkbeck College, University of London, in 1949. It was published two years later; see(Robinson 1951).

57. (Kochen 1976, 313).

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